In a survey conducted by a social worker 30 of the women res

In a survey conducted by a social worker, 30% of the women responded that they were the victims of domestic violence. Assume that this percentage is true for all women in the United States. Using the binomial probability distribution, find the probability that the number of women in a random sample of 20 who were victims of domestic violence is
(a) at least 6   
(b) between 7 and 11 (inclusive)
(c) at most 8

Solution

a)

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    20      
p = the probability of a success =    0.3      
x = our critical value of successes =    6      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   5   ) =    0.416370829
          
Thus, the probability of at least   6   successes is  
          
P(at least   6   ) =    0.583629171 [answer]

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b)

Note that P(between x1 and x2) = P(at most x2) - P(at most x1 - 1)          
          
Here,          
          
x1 =    7      
x2 =    11      
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    20      
p = the probability of a success =    0.3      
          
Then          
          
P(at most    6   ) =    0.608009812
P(at most    11   ) =    0.994861838
          
Thus,          
          
P(between x1 and x2) =    0.386852026   [answer]

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c)

Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    20      
p = the probability of a success =    0.3      
x = the maximum number of successes =    8      
          
Then the cumulative probability is          
          
P(at most   8   ) =    0.886668537 [answer]